Intro to Domain and Range

Q: How do I remember which one is domain and which is range?

Alphabetical order: $d$ before $r$, and $x$ before $y$. Domain is the $x$-values (inputs), range is the $y$-values (outputs). Domain is read horizontally on a graph; range is read vertically.

Q: When do I use braces versus inequalities?

Braces list a discrete set — separate points, like $\{1, 2, 4\}$. Inequalities or interval notation describe a continuous stretch of values, like $-1 \leq x < 3$ or $[-1, 3)$. Use braces for dots, inequalities for unbroken curves.

Q: Can the domain or range be all real numbers?

Yes, and for lines it usually is: a non-horizontal line like $y = 2x + 1$ has both domain and range equal to all real numbers. Curves with a highest or lowest point, like parabolas and absolute value graphs, keep the full domain but have a restricted range.

The domain of a function is the set of all possible inputs — every $x$ -value the function uses. The range is the set of all possible outputs — every $y$ -value the function produces. That's the whole definition, and every domain-and-range question is just that definition applied to points, a graph, or an equation.

The easiest way to keep them straight: in an ordered pair $(x, y)$ , the $x$ comes first alphabetically and domain comes before range alphabetically. Domain goes with $x$ , range goes with $y$ . Everything else in this topic is reading carefully.

Domain and range of a set of points

When a relation is a list of ordered pairs, the domain is the set of first coordinates and the range is the set of second coordinates. For $\{(1, 4), (2, 7), (3, 10)\}$ , the domain is $\{1, 2, 3\}$ and the range is $\{4, 7, 10\}$ .

Two housekeeping rules: list each value only once, even if it repeats, and write the values in increasing order. For $\{(2, 5), (4, 5), (6, 8)\}$ , the range is $\{5, 8\}$ — the repeated $5$ is listed a single time.

The mapping diagram below shows the relation $\{(1, 4), (2, 7), (3, 10)\}$ : the left oval collects the domain (the inputs) and the right oval collects the range (the outputs).

Reading domain and range from a graph

For the domain, scan the graph left to right and ask: which $x$ -values does the graph sit above or below? For the range, scan bottom to top and ask: which $y$ -values does the graph reach? Domain is a horizontal question; range is a vertical question.

Arrows matter. An arrow means the graph continues forever in that direction, so that end of the domain or range is unbounded. Endpoints matter too: a filled circle means the value is included (use $\leq$ or a square bracket), while a hollow circle means it is not (use $<$ or a round bracket).

The parabola below continues forever left, right, and upward, but it has a lowest point at $(1, -2)$ . So its domain is all real numbers, while its range is only $y \geq -2$ — no point on the curve sits below the vertex.

Continuous graphs: inequalities and intervals

A discrete graph — separate dots — gets a list in braces. A continuous graph — an unbroken curve or segment — gets described with inequalities or interval notation, because it covers every value between its endpoints, not just the whole numbers.

A segment that runs from $x = -1$ (filled) to $x = 3$ (hollow) has domain $-1 \leq x < 3$ , or $[-1, 3)$ in interval notation. Square bracket for included, round bracket for excluded — the brackets are just the circle types translated into symbols.

Worked examples

Example 1: a set of ordered pairs

Find the domain and range of $\{(1, 3), (2, 5), (4, 9)\}$ .

Collect the first coordinates

1, 2, 4

That set is the domain

\{1, 2, 4\}

Collect the second coordinates

3, 5, 9

That set is the range

\{3, 5, 9\}

Answer: domain $\{1, 2, 4\}$ , range $\{3, 5, 9\}$

Example 2: a segment with endpoints

A segment runs from $(-1, 2)$ to $(3, 4)$ , with both endpoints filled in. Find the domain and range.

Scan left to right:

x

runs from

-1

3

, both included

-1 \leq x \leq 3

Scan bottom to top:

y

runs from

2

4

, both included

2 \leq y \leq 4

In interval notation

[-1, 3] \text{ and } [2, 4]

Answer: domain $-1 \leq x \leq 3$ , range $2 \leq y \leq 4$

Example 3: a parabola

Find the domain and range of $y = x^2 - 4$ .

The graph extends forever left and right

\text{domain: all real numbers}

Find the lowest point — the vertex

(0, -4)

The parabola opens upward from there, so

y

never drops below

-4

y \geq -4

Answer: domain: all real numbers; range: $y \geq -4$

Example 4: an absolute value function

Find the domain and range of $y = |x - 2| + 1$ .

Any

x

can be substituted

\text{domain: all real numbers}

The smallest

|x - 2|

can be is

0

, at

x = 2

y = 0 + 1 = 1

Every other input gives something larger

y \geq 1

Answer: domain: all real numbers; range: $y \geq 1$

Try one yourself

Common questions

How do I remember which one is domain and which is range?

Alphabetical order: $d$ before $r$ , and $x$ before $y$ . Domain is the $x$ -values (inputs), range is the $y$ -values (outputs). Domain is read horizontally on a graph; range is read vertically.

When do I use braces versus inequalities?

Braces list a discrete set — separate points, like $\{1, 2, 4\}$ . Inequalities or interval notation describe a continuous stretch of values, like $-1 \leq x < 3$ or $[-1, 3)$ . Use braces for dots, inequalities for unbroken curves.

Can the domain or range be all real numbers?

Yes, and for lines it usually is: a non-horizontal line like $y = 2x + 1$ has both domain and range equal to all real numbers. Curves with a highest or lowest point, like parabolas and absolute value graphs, keep the full domain but have a restricted range.

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